Can you explain again how 2's complement is useful?

If negative numbers are encoded as 2's complement, addition and subtraction works much more easily. You can add 2's-complement-encoded numbers with the usually binary addition method, whereas the signed-magnitude method of encoding negative numbers (for which the leftmost bit is assigned to carry the minus sign information) is not compatible with that.

Take the example from class:

The 2's complement version of $(63)_{10}$ is 00111111, and of $(-63)_{10}$ is 11000001. Now add these (by the method in the question above, i.e., in each place ``carry'' a resulting 1 to the next-most-significant bit, and continue right to left). This yields 00000000, which is what you expect from adding $63$ and $-63$.

Now suppose you try the same with signed-magnitude-encoded numbers. $63 + -63$ in signed-magnitude binary is $00111111 + 10111111$, which by binary addition yields $11111110$, which is not zero. So adding these requires some tricky manipulation.

(We'll see how to implement the carry-the-overflow-bit method of binary addition in digital electronics a couple of lectures from now.)